Question: What is $3^{-1} + 3^{-2} \pmod{25}$? Express your answer as an integer from $0$ to $24$, inclusive.
Solution: If we first get a "common denominator" as if 3 and 9 represent real numbers rather than residues, we get $$\frac 13 + \frac 19 \equiv \frac{9 + 3}{27} \equiv \frac{12}{2} \equiv \boxed{6} \pmod{25}.$$We can justify this as follows: let $a \equiv 3^{-1} \pmod{25}$ and $b \equiv 9^{-1} \pmod{25}$. Then $27a \equiv 9 \pmod{25}$ and $27b \equiv 3 \pmod{25}$. Summing these congruences shows that $27(a+b) \equiv 2(a+b) \equiv 9 + 3 \equiv 12 \pmod{25}$, so $a+b \equiv 6 \pmod{25}$, as desired.